This is the course book for MAP101. The presentation in Elk River, MN and Chicago, IL are based on this.

1. Singapore Mathematics:
An Introduction
What is Singapore Mathematics?
There is no such thing as Singapore Mathematics in Singapore. What
has come to be known as Singapore Mathematics is the way students
learn Mathematics and the way teachers learn to teach Mathematics
in Singapore. This includes the curriculum, the textbooks, and the
corresponding teacher professional development.
The Singapore Mathematics curriculum is derived from an education
system that focuses on thinking and places a strong emphasis on
conceptual understanding and mathematical problem-solving. The
scope and sequence of the curriculum is well articulated and follows a
spiral progression. A pedagogy that is based on students progressing
from concrete to pictorial and then to abstract representations,
helps the majority of students acquire conceptual understanding of
mathematical concepts. Visuals are used extensively in textbooks.
Mathematics as a Vehicle to
Develop Thinking Skills
Since the late 1990s, the Singapore education system has emphasized
on thinking skills as one of its pillars. Schools are encouraged to use
school subjects to help students to acquire good thinking skills and
develop good thinking habits. In this vein, the ‘Thinking Schools,
Learning Nation’ philosophy was introduced in 1997.
The latest version of the Singapore Mathematics (Primary) syllabus
states that Mathematics is “an excellent vehicle for the development
and improvement of a person’s intellectual competence in logical
reasoning, spatial visualisation, analysis and abstract thought” (Ministry
of Education of Singapore, 2006, p. 5).
The following Lesson 1 helps students develop visualization ability.
Lesson 2 helps students develop the ability to see patterns and
generalize the patterns. Visualization and generalization are examples of
intellectual competence that can be developed through Mathematics.
1

2. Lesson 1 In the Kitchen
Allan puts some brown sugar on a dish.
The total weight of the brown sugar and dish is 110 g.
Bella puts thrice the amount of brown sugar that Allan puts
on an identical dish, and the total weight of the brown
sugar and dish is 290 g.
Find the weight of the brown sugar that Bella puts on the dish.
2

3. Lesson 2 Name Patterns
Cheryl and David tried counting the letters in their names in a certain way.
For example, Cheryl counted the letters in CHERYL back and forth such
that the letter C is the 1st, H is the 2nd, E is the 3rd, R is the 4th, Y is the 5th,
and L is the 6th. Then she counts backwards such that Y is the 7th, R is the 8th,
E is the 9th, H is the 10th, C is the 11th. Using this way of counting, Cheryl’s
19th letter is E. When David counts his name DAVID, the 19th letter is V.
Find Cheryl’s and David’s 99th letter using this way of counting.
3

4. Mathematical Problem Solving
as the Focus of Learning
Mathematics
Based on research from around the world, Singapore developed a
Mathematics curriculum in the late 1980s to enable students to develop
mathematical problem-solving ability. This was introduced to Primary 1
students in 1992.
Beliefs
Interest
Appreciation
Monitoring of one’s own thinking
Confidence M
et Self-regulation of learning
Perseverance
es ac
d og
tu nit
A tti ion
Mathematical
s
Problem
esse
Numerical calculation Reasoning, communication
Skill
Algebraic manipulation Solving and connections
Proc
Thinking skills and heuristics
s
Spatial visualization
Application and modelling
Data analysis
Measurement Concepts
Use of mathematical tools
Estimation Numerical
Algebraic
Geometrical
Statistical
Probabilistic
Analytic
In Lessons 3, 4, and 5, students do not have rules that they can
follow to solve the problems. Thus, students have to think of ways
to solve the problems and in this way, develop problem-solving
strategies. Lesson 3 helps to teach students a new skill. In Lesson
4, the problem is used to consolidate a skill, by allowing students to
practice. In Lesson 5, students have to apply a previously learned skill
to solve a problem involving the formula for a figure that is different
from the ones they have been taught, as students are not taught the
formula to find the area of a trapezium (also known as trapezoid) in the
Singapore primary curriculum.
4

5. Lesson 3 Sharing Three-Quarters
Share three-quarters of a cake equally among 4 persons.
What fraction of the cake does each person get?
5

6. Lesson 4 Make a Multiplication Sentence
Use one set of digit tiles to make correct multiplication sentences.
×
Resources: Digit Tiles (See Appendix)
6

7. Lesson 5 What is the Area?
Find the area of the figure.
5 cm
4 cm 5 cm
8 cm
7

8. Learning Theories
A strong foundation is necessary for the students to do well in Mathematics.
In the Singapore textbooks, such a strong foundation is achieved through
the application of a few learning principles or learning theories.
Jerome Bruner
The Concrete Pictorial Abstract Approach — the progression
from concrete objects to pictures to abstract symbols is recommended
for concept development. This is based on the work of Jerome Bruner
on enactive, iconic, and symbolic representations.
Students learn a new concept or skill by using concrete materials.
Bruner referred to this as the enactive representations of the concept
or skill. Later, pictorial representations are used before the introduction
of symbols (abstract representations). Reinforcement is achieved by
going back and forth between the representations.
For example, students learn the concept of division by sharing 12
cookies among 4 persons as well as by putting 12 eggs in groups of 4
before progressing to using drawings to solve division problems. Later,
they learn to write the division sentence 12 ÷ 4 = 3.
This is referred to as the CPA Approach.
Let ’s Learn!
15 Division
How To Divide
Sharing
Let ’s Learn!
1
Sharing Equally
Googol has 6 mangosteens.
1 There are 12 cookies. He wants to divide the mangosteens into 2 equal groups.
Googol has 4 friends. How many mangosteens are there in each group?
He gives each friend the same
number of cookies in a bag. ÷ stands for
division.
6÷2 =3
Indu Huiling Amil Weiming There are 3 mangosteens in each group.
I try putting 2 cookies Now he wants to divide them into equal 3 groups.
in each bag. Then I
have 4 cookies left. 6÷3 =2
There are 2 mangosteens in each group.
How do I read
6 ÷ 2 = 3 and 6 ÷ 3 = 2 are division sentences. 6 ÷ 3 = 2?
We read 6 ÷ 2 = 3 as six divided by two
Now I put 1 more cookie is equal to three.
in each bag. I have no
Each friend gets 3 cookies.
cookies left.
79 81
079-083 MthsP1B U15.indd 79 6/27/06 6:14:37 PM MPH!Mths 2A U04.indd 81 7/7/06 8:25:15 PM
Pupil’s Book 1B p. 79 Pupil’s Book 2A p. 81
8

9. Grouping
Let ’s Learn! 3 Googol has 12 snap cards.
He divides the cards equally among his friends.
Finding The Number Of Groups First put 4 eggs Each friend gets 4 cards. How many friends are there?
1 There are 12 eggs. into 1 bowl.
Put 4 eggs into each bowl.
How many bowls do you need?
Do this until all the eggs × 4 = 12
are put into the bowls.
12 ÷ 4 =
There are friends.
4 Sulin has 18 cards.
She gives the cards to some friends.
I need 3 bowls.
If each friend gets 3 cards, how many friends are there?
2 Crystal has 15 toy cats.
She puts 3 toy cats on each sofa.
How many sofas are needed for all the toy cats?
× 3 = 18
18 ÷ 3 =
sofas are needed for all the toy cats. There are friends.
81 83
079-083 MthsP1B U15.indd 81 6/27/06 6:15:11 PM
Pupil’s Book 1B p. 81 Pupil’s Book 2A p. 83
MPH!Mths 2A U04.indd 83 1/25/07 9:30:34 AM
Source: My Pals are Here! Maths (2nd Edition)
The Spiral Approach — students revisit core ideas as they deepen their
understanding of those ideas. This is also one of Jerome Bruner’s theories.
For example, students learn to divide discrete quantities without the
need to write division sentences in Primary 1.
I put 15 apples equally
on 3 plates.
I put 3 apples in a group.
Divide 15 apples into 3 equal groups. Divide 15 apples into groups of 3.
There are apples in each group. There are groups.
62
61
PFP_1BTB_Chpt15.indd 62 2/2/07 4:41:06 PM
Textbook 1B p. 61 Textbook 1B p. 62
PFP_1BTB_Chpt15.indd 61 2/2/07 4:41:01 PM
In Primary 2, they revisit this idea and use division sentences to
represent the word problems.
4.
2 Division 1.
Divide 8 mangoes into 2 equal groups. Divide 15 children into groups of 5.
whole There are 4 mangoes in each group. There are 3 groups.
We write: Divide 8 by 2. We write:
The answer is 4.
8 ÷ 2=4 15 ÷ 5 = 3 Divide 15 by 5.
The answer is 3.
part part part
We also divide to find
This is division. the number of groups.
Divide 12 balloons
into groups of 4. We divide to find the
number in each group.
Divide 12 balloons into
3 equal groups.
97 Exercise 5, pages 100-102
94 95
Textbook 2A p.97
PFP_2A_TB_Chap5.indd 97 2/2/07 3:25:51 PM
Textbook 2A p.94 Textbook 2A p.95
PFP_2A_TB_Chap5.indd 94 2/2/07 3:25:40 PM PFP_2A_TB_Chap5.indd 95 2/2/07 3:25:43 PM
Source: Primary Mathematics (Standards Edition)
9

10. In Primary 3, the idea is extended to include the idea of a remainder.
They also learn to regroup before dividing 2-digit and 3-digit numbers.
2 4 furries shared 11 seashells equally among themselves.
Let ’s Learn!
Division With Regrouping In Tens And Ones
1 Fandi and Farley went fishing and caught
some fishes and crabs.
They shared the 52 fishes equally between
themselves.
How many fishes did each boy get?
a How many seashells did each furry receive?
b How many seashells were left? 52 ÷ 2 = ?
First, divide the tens by 2. 2
Tens Ones
5 tens ÷ 2 2 ͤෆෆ
5 2
a 11 ÷ 4 = ? = 2 tens with remainder 1 ten 4
Divide the 1
11 seashells into
4 equal groups.
4×2=8
8 is less than 11.
4 × 3 = 12
12 is more than 11. Regroup the remainder ten: 2
Tens Ones
Choose 2. 1 ten = 10 ones 2 ͤෆෆ
5 2
Add the ones: 4
10 ones + 2 ones = 12 ones 1 2
11 ones ÷ 4 = 2 ones with remainder 3 ones
= 2 R3
Quotient = 2 ones 2 R3
Then, divide the ones by 2. 2 6
Remainder = 3 ones 4 ͤෆෆ
1 1 Tens Ones
12 ones ÷ 2 = 6 ones 2 ͤෆෆ
5 2
8 4
Each furry received 2 seashells. 3 1 2
So, 52 ÷ 2 = 26. 1 2
b 3 seashells were left. 0
Each boy got 26 fishes.
94 101
MthsP3A_U07(1 July) 94 7/5/06 2:39:28 PM
MthsP3A_U07(1 July) 101 7/5/06 2:40:54 PM
Pupil’s Book 3A p. 81 Pupil’s Book 3A p. 101
In Primary 4, 4-digit numbers are used and in Primary 5, division of
continuous quantities are dealt with where 168 ÷ 16 = 10.5 rather than
10 remainder 8.
Division
Let’s Learn!
7 a Divide 4572 by 36. Press Display
C 0
Division By A 1-Digit Number 4572 4572
1 6381 sweets were given to the children at a fun fair. Each child received ÷ 36 36
3 sweets. How many children were there at the fun fair?
= 127
Th H T O The answer is 127.
Step 1
2
Divide 6 thousands by 3.
3 6 3 8 1 b
6 2؋3 What is 168 � divided by 16? Press Display
6 thousands ، 3 = 2 thousands
C 0
= 2000
Step 2 168 168
2 1
3 6 3 8 1 ÷ 16 16
Divide 3 hundreds by 3.
6
= 10.5
3 hundreds ، 3 = 1 hundred 3 168 � divided by 16 is 10.5 �.
= 100 3 1؋3
Step 3 2 1 2
3 6 3 8 1
Divide 8 tens by 3. 6 8 Carry out this activity.
3 Remember to press
8 tens ، 3 = 2 tens with remainder 2 tens 3 C before you start
= 20 with remainder 20 Work in pairs to do these: working on each sum.
8
6 2؋3
a 1065 ؋ 97 b 13 674 ؋ 7 c 1075 ، 25
2
d 10 840 ، 40 e 25 m ؋ 48 m f 406 g ، 28
Step 4 2 1 2 7
Divide 21 ones by 3. 3 6 3 8 1 Think of one multiplication and one division sentence. Get your
6 partner to work them out using a calculator. Check that your
21 ones ، 3 = 7 ones 3 partner’s answers are correct using your calculator.
3
=7
8
6 WB 5A, p 23
When 6381 is divided by 3, the quotient Practice 1
2 1
is 2127 and the remainder is 0.
2 1 7؋3
There were 2127 children at the fun fair.
0
57 32 Chapter 2: Whole Numbers (2)
J43 4A CB U03 (57-70) 28Jul 57 7/28/06 2:56:07 PM
Pupil’s Book 4A p. 57 Pupil’s Book 5A p. 32
CB5A_U02(29-42).indd 32 9/7/07 10:26:28 AM
Source: My Pals are Here! Maths (2nd Edition)
Jerome Bruner proposed the idea of spiral curriculum in 1960s.
A curriculum as it develops should revisit [the] basic ideas repeatedly,
building upon them until the student has grasped the full formal
apparatus that goes with them.
Bruner, 1960
Bruner recommended that when students first learn an idea, the
emphasis should be on grasping the idea intuitively. After that, the
curriculum should revisit the basic idea repeatedly, each time adding
to what the students already know until they understand the idea fully.
Bruner emphasized that ideas are not merely repeated but… revisited
later with greater precision and power until students achieve the reward
of mastery.
Bruner, 1979
10

11. Bruner explained that ideas that have been introduced in an intuitive
manner were then revisited and reconstrued in a more formal or
operational way, then being connected with other knowledge, the
mastery at this stage then being carried one step higher to a new level
of formal or operational rigour and to a broader level of abstraction
and comprehensiveness. The end stage of this process was eventual
mastery of the connexity and structure of a large body of knowledge.
Bruner, 1960
Bruner gave an example of how students learn the idea of prime
numbers and factoring.
The concept of prime numbers appears to be more readily grasped
when the child, through construction, discovers that certain handfuls
of beans cannot be laid out in completed rows and columns. Such
quantities have either to be laid out in a single file or in an incomplete
row-column design in which there is always one extra or one too few
to fill the pattern. These patterns, the child learns, happen to be called
prime.
Bruner, 1973
Zoltan Dienes
Systematic Variation – Students are presented with a variety of tasks
in a systematic way. This is based on the ideas of Zoltan Dienes
(Dienes, 1960).
Let ’s Learn!
2 Addition And Subtraction
Within 1000 Addition With Regrouping In Ones
1 347 + 129 = ?
Let’s Learn! First, add the ones.
Hundreds Tens Ones
3 14 7
Simple Addition Within 1000 + 1 29
1 Add using base ten blocks. First, add the ones. 347 6
Use the place value chart to help you.
1 23 7 ones + 9 ones
a 123 + 5 = ? + 5 = 16 ones
8 Regroup the ones.
Hundreds Tens Ones 16 ones = 1 ten 6 ones
3 ones + 5 ones 129
= 8 ones Then, add the tens.
Then, add the tens. 3 14 7
123
+ 1 29
1 23
+ 5 Hundreds Tens Ones 7 6
4 tens + 2 tens + 1 ten
28
5 = 7 tens
2 tens + 0 tens
= 2 tens Lastly, add the
hundreds.
Lastly, add the 476
So, 123 + 5 = 128. 3 14 7
hundreds.
+ 1 29
1 23
+ 5 47 6
3 hundreds + 1 hundred
1 28 So, 347 + 129 = 476. = 4 hundreds
1 hundred + 0 hundreds
= 1 hundred
27 35
MPH!Mths 2A U02.indd 27 7/7/06 7:43:55 PM MPH!Mths 2A U02.indd 35 7/7/06 7:46:27 PM
Pupil’s Book 2A p. 27 Pupil’s Book 2A p. 35
Source: My Pals are Here! Maths (2nd Edition)
The above example shows mathematical variability. The variation is in
the Mathematics — addition without regrouping and with regrouping.
11

12. Multiplying by a
Les
1-Digit Number
The next example shows perceptual variability — the mathematical
concept is the same but students are presented with different ways to
perceive a 2-digit number.
Lesson Objective
• Use different methods to multiply up to 4-digit numbers
by 1-digit numbers, with or without regrouping.
arn Represent numbers using place-value charts.
Le
213 can be represented in these ways.
nTextbook 1B p. 30
o
Less Multiplying by a
Source: Primary Mathematics (Standards Edition)
Hundreds multiple embodiment is to use different ways to
Tens Ones
The idea of
1-Digit Numbera
represent the same concept. In the above example, the concept of
2-digit number Lesson Objective
such as 34 is represented in multiple ways — using
sticks, coins and base ten blocks. Is the representation more abstract
• Use different methods to multiply up to 4-digit numbers
than another? by 1-digit numbers, with or without regrouping.
In the next example, 3-digit numbers are represented using base ten
arn Represent numbers using place-value charts.
Le
blocks, number discs and digits.
213 can be represented in these ways.
Multiplying by a
n
o
Less
Lesson Objective
1-Digit Number
Hundreds
• Use different methods to multiply up to 4-digit numbers
Tens Ones
by 1-digit numbers, with or without regrouping.
arn Represent numbers using place-value charts.
Le
213 can be represented in these ways.
Hundreds Tens Ones
Hundreds Tens Ones
Hundreds Tens Ones
Hundreds Tens Ones
2 1 3
Lesson 3.1 Multiplying by a 1-Digit Number 77
Student Book 4A
Source: Math in Focus: The Singapore Approach
Hundreds Tens Ones
The base ten blocks are shown proportionately using concrete
materials. For example, looks ten times as large as .
However, the number discs are non-proportionate.
Hundreds
For example, Hundreds Tens
does not look ten times as large as . Tens OnesOnes
2 1 3
2
This is another example of representing the same mathematical
concept in different ways, some are more abstract than others. It is im-
1 3
portant to provide students with these variations in a systematic way.
Additional reading:
Read an article by Post (1988) which has a section on Dienes’ ideas of variability (http://www.cehd.
umn.edu/rationalnumberproject/88_9.html), as well as the six-stage theory of learning mathematics 3.1
Lesson Multiplying by a 1-Digit Nu
(http://www.zoltandienes.com/?page_id=226)
12

13. Richard Skemp
Richard Skemp (Skemp, 1976) provides Mathematics teachers with a
way to think about what constitutes understanding in Mathematics.
Skemp distinguished between the ability to perform a procedure,
(for example, long division), and the ability to explain the procedure,
(for example, explaining the rationale for ‘invert-and-multiply’ when
dividing a proper fraction by a proper fraction). He refers to the former
as instrumental understanding (or procedural understanding) and the
latter as relational understanding (conceptual understanding).
Singapore Mathematics curriculum expects instrumental understanding
to be accompanied by relational understanding. It is pointless to learn
a procedure without having a conceptual understanding.
“Although students should become competent in the various
mathematical skills, over-emphasising procedural skills without
understanding the underlying mathematical principles should be
avoided” (Ministry of Education of Singapore, 2006, p. 7).
Conventional understanding involves the ability to understand the use
of conventions. For example, it is a convention to use + as the symbol
for addition. Some conventions are not universal. For example, ÷ is
used as the symbol for division in some countries, but : is used as the
symbol for division in others. Conventions that are universal include
the order of operations. There are some facts, names, notations, and
usage which are universally agreed upon, and there are no particular
reasons for using those conventions.
12. Chelsea has 5 apple tarts. She cuts each tart into 1 . Find the number
2
of half-tarts Chelsea has.
5 ÷ 1 = 10 When you cut a whole into
2 halves, you get 2 halves.
So, in 5 wholes there are
Chelsea has half-tarts. 5 × 2 halves.
13. 3 cakes are shared equally among some children. Each child gets
3 of a cake. How many children got 3 of a cake?
4 4
When you cut a whole into three-quarters,
you get 1 1 three-quarters. So, in 3 wholes
3
3÷ 3 =4 there are 3 × 1 1 three-quarters.
4 3
children shared the cakes.
66
Textbook 6A
6ATB_Unit 3.indd 66 4/24/09 5:10:13 PM
Source: Primary Mathematics (Standards Edition)
Additional reading:
Read the classic article originally published in Mathematics Teaching (1976) at http://www.grahamtall.
co.uk/skemp/pdfs/instrumental-relational.pdf
13

14. Lesson 6 Long Division
Find the value of 51 ÷ 3.
In some countries, this can also be written as 51 : 3.
51
14

15. Lesson 7 Bar Model
1 1
Marcus gave of his coin collection to his sister and of the remainder
4 2
to his brother.
As a result, Marcus had 18 coins.
Find the number of coins in his collection at first.
In Lessons 6 and 7, we are able to explain the long division algorithms
as well as the procedure to multiply fractions. We are said to possess
relational understanding of these procedures.
15

16. References
1. Bruner, J. (1960). The Process of Education. Cambridge, MA: Harvard
University Press.
2. Bruner, J (1966). On Knowing: Essays for the Left Hand. Cambridge,
MA: Harvard University Press.
3. Bruner, J. S. (1973). Beyond the information given: Studies in the
psychology of knowing, pp. 218-238. New York: W. W. Norton
& Co Inc.
4. Dienes, Z. P. (1960). Building up mathematics. London: Hutchinson
Educational Ltd.
5. Ministry of Education of Singapore. (2006). Mathematics Syllabus
(Primary). Singapore: Curriculum Planning and Development Division.
from http://www.moe.gov.sg/education/syllabuses/sciences/files/
maths-primary-2007.pdf
6. Post, T. (1988). Some notes on the nature of mathematics learning.
Teaching Mathematics in Grades K-8: Research Based Methods ,
pp. 1-19. Boston: Allyn & Bacon.
http://www.cehd.umn.edu/rationalnumberproject/88_9.html
7. Skemp, R. R. (1976). Relational and instrumental understanding.
Mathematics Teaching, 77, pp. 20-26.
http://www.grahamtall.co.uk/skemp/pdfs/instrumental-relational.pdf
16

17. Appendix
0 1 0 1
2 3 2 3
4 5 4 5
6 7 6 7
8 9 8 9
17

18. Notes
18